1.2 Domain and Range

Domain and Range of Functions

Domain and range define the boundaries of a function: what you're allowed to put in, and what you can get out. Mastering these concepts is essential because nearly every topic in pre-calculus (transformations, compositions, inverses) depends on knowing which inputs and outputs are valid.

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Domain of Equation-Defined Functions

The domain is the set of all possible input values (x-values) for a function. To find it, start by assuming all real numbers work, then look for restrictions that force you to exclude certain values.

There are two main algebraic restrictions to watch for:

1. Division by zero is undefined. If your function has a variable in the denominator, set the denominator not equal to zero and solve.

• For a rational function like $\frac{f(x)}{g(x)}$, solve $g(x) = 0$ to find the x-values you must exclude from the domain.
• Example: For $h(x) = \frac{3}{x - 4}$, set $x - 4 = 0$, so $x = 4$ is excluded. The domain is all real numbers except 4.

2. Even-indexed roots require non-negative radicands. Square roots, fourth roots, and so on are only defined when the expression inside is $\geq 0$.

• For a function like $f(x) = \sqrt{2x + 6}$, solve $2x + 6 \geq 0$, giving $x \geq -3$. The domain is $[-3, \infty)$.
• Odd-indexed roots (cube root, fifth root, etc.) accept all real numbers, so they don't restrict the domain.

When both restrictions appear in the same function, you need to satisfy all of them simultaneously.

Real-world context can add further limits:

• Physical quantities like length, distance, or time are typically non-negative.
• Discrete quantities like the number of people or items must be whole numbers.

These contextual restrictions override the algebraic domain. A function might be algebraically defined for all $x > 0$, but if $x$ represents the number of students, only positive integers make sense.

Piecewise Functions and Their Properties

A piecewise function is defined by different equations on different intervals of the domain. Each "piece" has its own rule and its own interval.

To graph a piecewise function:

1. Identify each interval and its corresponding equation.
2. Graph each piece only on its specified interval.
3. Mark endpoints carefully: use a closed circle (●) if the endpoint is included ($\leq$ or $\geq$) and an open circle (○) if it's excluded ($<$ or $>$).

At a boundary between two pieces, only one piece should "own" that x-value. Check that exactly one piece includes each boundary point; otherwise the relation isn't a function at that point.

• The domain of a piecewise function is the union of all the individual intervals.
• The range is the union of all the y-values produced by each piece on its interval. You need to evaluate each piece's behavior (its minimum and maximum output) on its specific interval, then combine the results.

Notation for Domain and Range

You'll use two main notations throughout this course.

Interval notation uses parentheses, brackets, and the infinity symbol:

• Parentheses $(\ )$ mean the endpoint is not included (open).
• Brackets $[\ ]$ mean the endpoint is included (closed).
• Infinity ($\infty$) and negative infinity ($-\infty$) always get parentheses because they aren't actual numbers you can reach.
• The union symbol $\cup$ joins separate intervals. For example, the domain "all reals except 4" is written $(-\infty, 4) \cup (4, \infty)$.
• Example: $(2, 5]$ means all numbers greater than 2 and up to and including 5.

Set-builder notation describes a set using a variable and a condition:

• General form: $\{x \mid \text{condition}\}$, read as "the set of all x such that the condition is true."
• Example: $\{x \mid x \in \mathbb{R},\ x > 0\}$ means all positive real numbers.
• Example: $\{x \mid x \in \mathbb{Z},\ -3 \leq x < 5\}$ means all integers from $-3$ to $4$, inclusive.

Both notations convey the same information. Interval notation is more compact and tends to be used more often in pre-calculus, but set-builder notation is useful when the condition is complex or involves specific number sets like integers.

Function Relationships and Properties

• A continuous function has no breaks, gaps, or jumps in its graph. You could trace it without lifting your pencil. Continuity matters for domain and range because a continuous function on a closed interval $[a, b]$ will hit every y-value between $f(a)$ and $f(b)$.
• The codomain is the set of all possible output values a function could theoretically produce. The range (also called the image) is the set of outputs the function actually produces. The range is always a subset of the codomain.
• Mapping refers to how a function assigns each input to exactly one output. This is the core requirement for a relation to qualify as a function.
• Inverse functions reverse the input-output relationship: if $f(a) = b$, then $f^{-1}(b) = a$. The domain of $f$ becomes the range of $f^{-1}$, and vice versa. This swap is why understanding domain and range now pays off when you study inverses later.